Problem:  Jonna brought home a strip of wood 120 cm in length and 2 cm in width.  She instructed her daughter, Diana, to make her a rectangular picture frame and to use all of the wood.  She wanted the frame to be made as shown below.
 What are several sizes of frames Diana could make?
 What frame would give Diana the largest enclosed area?

    Frame Style:

Math Topic/Concept:  Area and perimeter.

Materials: Pencil, paper, graph paper, calculators (optional)

Classroom Use: (Developmental/Evaluation)

Classroom use comments*:  It will likely take some discussion to decide that the outside perimeter and the insider perimeter of the design is not 120.  The design is not the strongest design, but has been chosen so the problem is easier to solve.  Students may need to spend time thinking about how the boards need to be cut so that the rectangular frame can be made.

Grade:  4 - 6

Grade Cluster: (LateElem/MS-Jr.High)

Illinois Goal:  Goal 7 and Goal 6

Standard: 7A3b, 6B3a, 6B3b, 6C3a, 6C3b

Applied? (1-4):  2

Source:  original

Answer:  There are several rectangles that can be made. The rectangle with an outside perimeter of 32 cm and an inside perimeter of 28 cm gives the largest area for the picture.  This area is 576 cm2.

Strategies Listed:  Making an organized list.

Solution:  See answer.  It is important to note that the area on the inside of the frame is desired.  The width of the wood matters here as well.

Extensions or related problems*:  The length or the width of the board can be varied.  Different ways to make the frame could also be discussed.

Intended rubric or assessment method:  The following rubric could be used.
Exceeds:  Student finds a pattern to the length and width combinations and lists possible combinations. Clear explanation and/ or work to show understanding.  Finds the combination that makes the largest area and explains why it is the largest. May use decimals or fractions in the pattern and identify an infinite number of possibilities.

Meets: Student inds a pattern to the length and width combinations and lists combinations with perhaps a few errors. Finds the frame dimensions that make the largest area but does not provide justification.  Otherwise, explains work fairly well.

Does not meet: Student may find some patterns or sizes for the frame. Work contains major errors. May not find the rectangle with the largest area. Does not explain or gives poor explanations.

Write-up submitted by:  Melfried Olson

Problem:  If a He + He + Ha is 10 and a Ha + Ha + He is 11,
                   How much is a He + He + He?

Math Topic/Concept:  Algebra

Classroom Use: (Introductory)

Classroom use comments: Algebra unit after they have been introduced to symbolic representations and variables.

Grade:  6-8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  6 and 8

Standard:  6.C.3a   and  8.D.3a

Source:  Awesome Math Problems for Creative Thinking  (Creative Publications)
                 Grade 6-8   ISBN   0-7622-1285-3

Answer:  9

Strategies Listed:  Guess and check or Logical Reasoning

Solution:  Since He + He + Ha is 10 and Ha + Ha + He is 11, then the sum of He + He + Ha and Ha + Ha + He is 21.  That means 3 He=s and 3 Ha=s are 21, so He + Ha must be 21 divided by 3, or 7.  If He + Ha is 7, and He + He + Ha is 10 then the extra He must be a 3.  That means He + He + He is 3 x 3, or 9.

Extensions or related problems:  Have the student make up their own similar problem and share with a partner.

Write-up submitted by:  Jenni Robinson and Denise Mann

Problem:
 Use a calculator to keep track of your score.  Begin at start by entering 100 in your calculator.  Choose a path to move across or downward through the maze as your ‘pinball’ drops.  Do not retrace any segments of the maze.  You may not move upward.  The object of the game is to arrive at the finish at the bottom of the maze with the largest number possible.

 

Math Topic/Concept:  operations on decimal numbers

Materials:  calculator and maze, above

Classroom Use: (Introductory/Developmental)

Classroom use comments*:  Students may begin with the notion that multiplication produces a larger product, and division produces a smaller quotient.  This exploration should lead them to discover what happens when multiplying or dividing by decimal numbers greater than or less than one.

Grade:  6 - 8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  6, Number sense

Standard:  6C3

Applied? (1-4):  1

Source:  Texas Instruments, 2000: T3 Middle School Math Institute

Answer:  ~~

Strategies Listed:  guess and check, noticing patterns

Extensions or related problems*:  Students may also be told to arrive at the smallest result possible at the finish, or at the number closest to the 100 they began with at the start.

Intended rubric or assessment method:  Informal assessment: journal write: when does multiplication (or division) result in a product larger than either factor? smaller?

Write-up submitted by:  M.K. Robbins

Problem:  Rebecca is wrapping presents for people at the shopping mall.  She had 8 ½ yards of ribbon when she began.  She used 2 1/3 yards of ribbon to wrap one present.  She needs to wrap 2 more presents of the same size.  Does she have enough ribbon?

Math Topic/Concept:  Measurement, Fractions

Materials:  Optional:  Ribbon, yardstick

Classroom Use: (Developmental)

Classroom use comments*:  Use after you have introduced addition, subtraction, and multiplication of mixed numbers.

Grade:  7

Grade Cluster: (MS-Jr.High)

Illinois Goal:  6 and 7

Standard:  6.C.3a   and  7.A.3a

Applied? (1-4):  3

Source:  Hot words, Hot topics  (Creative Publications)   ISBN 0-7622-0629-2

Answer:  Yes

Strategies Listed: Computation

Solution:  2 1/3 x 3 = 7  or   2 1/3 + 2 1/3 + 2 1/3 = 7

Other solution methods (if any)*:  If they don’t compute they must explain how they physically did the measuring.

Extensions or related problems*:  How much paper is left over?  8½ - 7 = 1½
Related:  A ribbon is 8 yards long.  A dressmaker wants to make bows, each of which requires exactly ¾ foot of ribbon. With no waste how many bows can she make.  (32)

Intended rubric or assessment method:  Worksheet with related addition and subtraction problems of mixed numbers.  Set the problems as story problems.

Write-up submitted by:  Jenni Robinson and Denise Mann
 
 

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated Aug. 20, 2001